14.6. Definition of Cyclic Group

So now, let us define what we call as cyclic group. Let G be a group with some abstract operation ∘. It may or may not be a finite group. The specialty of the group is that it has an element g which we call as a generator. It is called a generator because when you take different powers of this generator, again by power I mean group exponentiation, you will get all the elements of your group. That means, this element g has the capacity to generate all the elements of your group by performing the group exponentiation on this generator. A group that has a generator g is called cyclic and is represented by the notation G = ⟨g⟩. This notation basically says that g can act as a seed and reproduce the entire set G by computing different powers of this generator.

A cyclic group is a special type of group where one element, known as the 'generator', can produce all the elements of the group by applying the group operation repeatedly. The notation ⟨g⟩ indicates that all members of the group can be found by raising g to various powers in the context of the group operation. For example, if g is a generator of a cyclic group, g^1, g^2, g^3, and so on would yield all the different elements in that group.

Imagine a group of people where there is a person, let's say Alice, who is the leader (the generator). Each time she invites another person to this gathering (which represents the group operation), all participants can be seen as producing different interactions or relationships within the group. Just like how Alice can bring in new members and create newer dynamics, the generator g can produce all the elements of the cyclic group.

Before proceeding further, let us see some examples of a cyclic group. So, consider the infinite group, namely the group based on the set of integers with respect to the plus operation. My claim is that the integer 1 constitutes your generator. This is because if you take different powers of this element 1, it will give you all the elements of your set of integers. So, let us see whether we can generate any arbitrary integer k by computing some power of this element k. And indeed, it is easy to verify that you take any integer k, it will be some n ⋅ 1 for some integer n.

This example illustrates an infinite cyclic group using integers with the addition operation. The integer 1 can generate all integers through multiplication by natural numbers. For instance, adding 1 repeatedly can produce 2, 3, and so on, while subtracting gets you the negatives. Therefore, in the context of cyclic groups, integer 1 serves to generate the whole set of integers.

Think of a bicycle wheel. The point where the tire touches the ground represents the integer 1 (the generator). Each complete rotation of the wheel adds to the distance covered (like adding integers) or moves the bicycle back in the opposite direction (subtraction). Just like this wheel can keep rolling indefinitely and cover all the ground, the generator in a cyclic group can create all elements by repeated application.

Now, let us take an example of a finite cyclic group. Let p be a prime and now if I consider the set of all integers modulo p, namely the set 0 to p − 1 and if my operation is + then my claim is that this group is a cyclic group and in fact has multiple generators. In fact, all the elements except the identity element 0 will be a generator for this group.

The example considers finite cyclic groups, particularly those formed by integers under addition modulo a prime number p. The set {0, 1, ..., p - 1} is viewed as a group, and all elements, except for zero, can serve as generators. This means that by adding these numbers to themselves (and taking mod p), you can generate every other number in the set, demonstrating a full cyclic structure.

Visualize a clock where the hour hand represents the group elements {0, 1, 2, ... , p−1}. If p is 12, we can think of adding hours: 1 o'clock leads to 2, 2 to 3, and so forth, cycling back after 12. Each hour (1 to 11 on the clock) can serve as a generator, effectively reaching the same 12-hour cycle, similar to how each element can generate the group.

So now, let us derive some interesting properties for cyclic groups. So, imagine G is a cyclic group and suppose the order of G is n. So that means, now I am considering a finite cyclic group since the group has a well-defined order. Let n be the number of elements and say g is one of the generators. Then my claim is that the order of the generator is n. What does that mean? This means that n is the smallest positive integer such that g^n is equal to the identity element.

In finite cyclic groups, the number of elements (order) is directly related to the generator's order as well. The generator g raised to the order n results in the identity element of the group. This concept emphasizes that the generator's repeated applications define the entire group's structure based on how many distinct elements are generated before returning to the identity.

Consider a musical note produced by an instrument as the generator. Each time you play that note consecutively (which represents applying the group operation), you form a melody until you return to the fundamental tone (the identity). This cycle continues, analogous to how a cyclic group operates until it returns to its fundamental state.

--